# Properties

 Label 2-63-7.2-c5-0-13 Degree $2$ Conductor $63$ Sign $-0.693 + 0.720i$ Analytic cond. $10.1041$ Root an. cond. $3.17870$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (3.19 − 5.53i)2-s + (−4.41 − 7.64i)4-s + (19.3 − 33.5i)5-s + (−87.5 − 95.6i)7-s + 148.·8-s + (−123. − 214. i)10-s + (−288. − 499. i)11-s + 391.·13-s + (−808. + 178. i)14-s + (614. − 1.06e3i)16-s + (−664. − 1.15e3i)17-s + (−471. + 816. i)19-s − 341.·20-s − 3.68e3·22-s + (−816. + 1.41e3i)23-s + ⋯
 L(s)  = 1 + (0.564 − 0.978i)2-s + (−0.137 − 0.238i)4-s + (0.346 − 0.599i)5-s + (−0.674 − 0.737i)7-s + 0.817·8-s + (−0.391 − 0.677i)10-s + (−0.718 − 1.24i)11-s + 0.642·13-s + (−1.10 + 0.243i)14-s + (0.599 − 1.03i)16-s + (−0.557 − 0.966i)17-s + (−0.299 + 0.518i)19-s − 0.191·20-s − 1.62·22-s + (−0.321 + 0.557i)23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $-0.693 + 0.720i$ Analytic conductor: $$10.1041$$ Root analytic conductor: $$3.17870$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{63} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 63,\ (\ :5/2),\ -0.693 + 0.720i)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.883186 - 2.07660i$$ $$L(\frac12)$$ $$\approx$$ $$0.883186 - 2.07660i$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1 + (87.5 + 95.6i)T$$
good2 $$1 + (-3.19 + 5.53i)T + (-16 - 27.7i)T^{2}$$
5 $$1 + (-19.3 + 33.5i)T + (-1.56e3 - 2.70e3i)T^{2}$$
11 $$1 + (288. + 499. i)T + (-8.05e4 + 1.39e5i)T^{2}$$
13 $$1 - 391.T + 3.71e5T^{2}$$
17 $$1 + (664. + 1.15e3i)T + (-7.09e5 + 1.22e6i)T^{2}$$
19 $$1 + (471. - 816. i)T + (-1.23e6 - 2.14e6i)T^{2}$$
23 $$1 + (816. - 1.41e3i)T + (-3.21e6 - 5.57e6i)T^{2}$$
29 $$1 - 1.46e3T + 2.05e7T^{2}$$
31 $$1 + (-1.95e3 - 3.38e3i)T + (-1.43e7 + 2.47e7i)T^{2}$$
37 $$1 + (-8.15e3 + 1.41e4i)T + (-3.46e7 - 6.00e7i)T^{2}$$
41 $$1 - 1.31e4T + 1.15e8T^{2}$$
43 $$1 - 1.47e4T + 1.47e8T^{2}$$
47 $$1 + (3.40e3 - 5.90e3i)T + (-1.14e8 - 1.98e8i)T^{2}$$
53 $$1 + (1.00e3 + 1.74e3i)T + (-2.09e8 + 3.62e8i)T^{2}$$
59 $$1 + (-2.57e4 - 4.45e4i)T + (-3.57e8 + 6.19e8i)T^{2}$$
61 $$1 + (2.05e4 - 3.55e4i)T + (-4.22e8 - 7.31e8i)T^{2}$$
67 $$1 + (2.52e4 + 4.38e4i)T + (-6.75e8 + 1.16e9i)T^{2}$$
71 $$1 + 3.99e4T + 1.80e9T^{2}$$
73 $$1 + (-2.78e4 - 4.82e4i)T + (-1.03e9 + 1.79e9i)T^{2}$$
79 $$1 + (-3.15e4 + 5.46e4i)T + (-1.53e9 - 2.66e9i)T^{2}$$
83 $$1 + 4.55e4T + 3.93e9T^{2}$$
89 $$1 + (-7.84e3 + 1.35e4i)T + (-2.79e9 - 4.83e9i)T^{2}$$
97 $$1 - 3.12e3T + 8.58e9T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−13.38044492524423633010194496177, −12.63170020607479222791478660120, −11.25998872055264165015616786521, −10.48094318499489890604593045445, −9.089910037857500450007260090837, −7.55263250614720839577557188872, −5.77488693490119691958478933538, −4.17474349886671882628330907561, −2.86225991646260963046018128783, −0.913659814833482489122263478614, 2.36558100513748689796042375319, 4.50870537155258379275123571692, 6.02826249314955159326179488984, 6.71873400840573116480909909272, 8.180035090900348004371191503094, 9.830360321556640579026806338965, 10.84297047278288871949512798150, 12.59956471139530015840316235422, 13.39574908955859247085733735743, 14.68151783989654694458387746094